Bifurcation Analysis of a Singular Bioeconomic Model with Allee Effect and Two Time Delays
A singular prey-predator model with time delays is formulated and analyzed. Allee effect is considered on the growth of the prey population. The singular prey-predator model is transformed into its normal form by using differential-algebraic system theory. We study its dynamics in terms of local ana...
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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2014/745296 |
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| _version_ | 1832567973653512192 |
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| author | Xue Zhang Qing-ling Zhang Zhongyi Xiang |
| author_facet | Xue Zhang Qing-ling Zhang Zhongyi Xiang |
| author_sort | Xue Zhang |
| collection | DOAJ |
| description | A singular prey-predator model with time delays is formulated and analyzed. Allee effect is considered on the growth of the prey population. The singular prey-predator model is transformed into its normal form by using differential-algebraic system theory. We study its dynamics in terms of local analysis and Hopf bifurcation. The existence of periodic solutions via Hopf bifurcation with respect to two delays is established. In particular, we study the direction of Hopf bifurcation and the stability of bifurcated periodic solutions by applying the normal form theory and the center manifold argument. Finally, numerical simulations are included supporting the theoretical analysis and displaying the complex dynamical behavior of the model outside the domain of stability. |
| format | Article |
| id | doaj-art-e707e61d5c094865a334ecd66b02441c |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-e707e61d5c094865a334ecd66b02441c2025-02-03T01:00:10ZengWileyAbstract and Applied Analysis1085-33751687-04092014-01-01201410.1155/2014/745296745296Bifurcation Analysis of a Singular Bioeconomic Model with Allee Effect and Two Time DelaysXue Zhang0Qing-ling Zhang1Zhongyi Xiang2Department of Science, Northeastern University, Shenyang, Liaoning 110819, ChinaDepartment of Science, Northeastern University, Shenyang, Liaoning 110819, ChinaDepartment of Mathematics, Hubei Institute for Nationalities, Enshi, Hubei 445000, ChinaA singular prey-predator model with time delays is formulated and analyzed. Allee effect is considered on the growth of the prey population. The singular prey-predator model is transformed into its normal form by using differential-algebraic system theory. We study its dynamics in terms of local analysis and Hopf bifurcation. The existence of periodic solutions via Hopf bifurcation with respect to two delays is established. In particular, we study the direction of Hopf bifurcation and the stability of bifurcated periodic solutions by applying the normal form theory and the center manifold argument. Finally, numerical simulations are included supporting the theoretical analysis and displaying the complex dynamical behavior of the model outside the domain of stability.http://dx.doi.org/10.1155/2014/745296 |
| spellingShingle | Xue Zhang Qing-ling Zhang Zhongyi Xiang Bifurcation Analysis of a Singular Bioeconomic Model with Allee Effect and Two Time Delays Abstract and Applied Analysis |
| title | Bifurcation Analysis of a Singular Bioeconomic Model with Allee Effect and Two Time Delays |
| title_full | Bifurcation Analysis of a Singular Bioeconomic Model with Allee Effect and Two Time Delays |
| title_fullStr | Bifurcation Analysis of a Singular Bioeconomic Model with Allee Effect and Two Time Delays |
| title_full_unstemmed | Bifurcation Analysis of a Singular Bioeconomic Model with Allee Effect and Two Time Delays |
| title_short | Bifurcation Analysis of a Singular Bioeconomic Model with Allee Effect and Two Time Delays |
| title_sort | bifurcation analysis of a singular bioeconomic model with allee effect and two time delays |
| url | http://dx.doi.org/10.1155/2014/745296 |
| work_keys_str_mv | AT xuezhang bifurcationanalysisofasingularbioeconomicmodelwithalleeeffectandtwotimedelays AT qinglingzhang bifurcationanalysisofasingularbioeconomicmodelwithalleeeffectandtwotimedelays AT zhongyixiang bifurcationanalysisofasingularbioeconomicmodelwithalleeeffectandtwotimedelays |